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12.2. The law of distribution of a linear function from an argument subject to the normal law

Lecture



Let the random variable   12.2.  The law of distribution of a linear function from an argument subject to the normal law subject to normal law with density:

  12.2.  The law of distribution of a linear function from an argument subject to the normal law , (12.2.1)

a random variable   12.2.  The law of distribution of a linear function from an argument subject to the normal law associated with her linear functional dependence:

  12.2.  The law of distribution of a linear function from an argument subject to the normal law . (12.2.2)

Where   12.2.  The law of distribution of a linear function from an argument subject to the normal law and   12.2.  The law of distribution of a linear function from an argument subject to the normal law - non-random coefficients.

It is required to find the distribution law   12.2.  The law of distribution of a linear function from an argument subject to the normal law .

Let's issue a solution in the form of two columns, similar to the example of the previous one.   12.2.  The law of distribution of a linear function from an argument subject to the normal law :

  12.2.  The law of distribution of a linear function from an argument subject to the normal law

  12.2.  The law of distribution of a linear function from an argument subject to the normal law

  12.2.  The law of distribution of a linear function from an argument subject to the normal law

  12.2.  The law of distribution of a linear function from an argument subject to the normal law

  12.2.  The law of distribution of a linear function from an argument subject to the normal law

  12.2.  The law of distribution of a linear function from an argument subject to the normal law

  12.2.  The law of distribution of a linear function from an argument subject to the normal law

  12.2.  The law of distribution of a linear function from an argument subject to the normal law

  12.2.  The law of distribution of a linear function from an argument subject to the normal law

  12.2.  The law of distribution of a linear function from an argument subject to the normal law

  12.2.  The law of distribution of a linear function from an argument subject to the normal law

  12.2.  The law of distribution of a linear function from an argument subject to the normal law

Transforming expression   12.2.  The law of distribution of a linear function from an argument subject to the normal law , we have:

  12.2.  The law of distribution of a linear function from an argument subject to the normal law ,

and this is nothing but a normal law with parameters:

  12.2.  The law of distribution of a linear function from an argument subject to the normal law (12.2.3)

If we go from standard deviations to proportional probable deviations, we get:

  12.2.  The law of distribution of a linear function from an argument subject to the normal law . (12.2.4)

Thus, we have seen that the linear function of the argument subject to the normal law is also subject to the normal law. To find the center of dispersion of this law, it is necessary to substitute its center of dispersion in the expression of a linear function instead of the argument. To find the standard deviation of this law, you need to multiply the standard deviation of the argument by the modulus of the coefficient for the argument in the expression of a linear function. The same rule holds true for probable deviations.


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Probability theory. Mathematical Statistics and Stochastic Analysis

Terms: Probability theory. Mathematical Statistics and Stochastic Analysis